[face=SPIonic]ou)k e)/xomen h(mei=j, w(/sper kai\ u(mei=j, oi)=kon me/giston[/face]Does this sentence mean: We do not have the biggest house, like you do.
Or does it mean: We, just like you, do not have the biggest house.

Is it possible for a word like [face=SPIonic]me/giston [/face]to be used to emphasise the size?
In other words, can it mean -very big- rather than -the biggest-?
In English this is quite common.

Yes, you can and probably should translate it as 'very big' here. If the adjective does not have an article even though it's in the superlative form then you often translate it like that (it's called elative translation).

As for the meaning of the sentence... well, I think that it must be you second translation. I would translate kai as 'aswell' here, 'and' would not make any sense. So they too have a small house.